##### 1. Square inequalities

The type inequality:

* аx² + b + c> 0 *or ах²

*ctively*

e

**+ bx + c <0 resp**e

* аx² + bx + c ≥* 0 or а

**x² + bx + c ≤ 0**where a,* b, c ar*e real numbers and a

**s called**

*≠ 0 i*

**a quadratic inequality with one variabl**e.Solve one of the quadratic inequalities* аx² + b + c> 0* or

*аx² + b + c <0*comes down to determining those values of x for* *which the corresponding quadratic function

*f (x) = аx² + bx + c*

gets a positive or negative value.

The set of solutions to the inequality

** аx² + bx + c ≥ 0** or

*аx² + bx + c ≤ 0*contains the zeros of the corresponding quadratic function

**Example.** Let us solve the inequali**ty х² – 2x -3 <0**

The corresponding quadratic function i** s f (x) = х² – 2х – **3. We sketch its graph with the help of its zeros, which are solutions of the equation

**. These are the number**

*х² – 2х – 3***a**

*s x₁ = -1*

*nd x₂ = 3*(Fig.1).

From the graph we see that it is negative, ie. the graph is bel*o*w the x-axis, for values of x from the interva**l (-1, 3***).*

##### 2. SYSTEM SQUARE INEQUALITIES

The general type of a system of quadratic inequalities with one variable is:

Instead of the ">*" s*ign in the inequalities it can be eithe

*r*<", or"

**'**'**≤ ", or"**

**≥**".

If M** Z **and

**Z are the sets of solutions of the first, ie the second inequality, then the set of solutions M o**

*M***the system is the intersection of those sets, ie.**

*f***M = M₁ ∩ M**₂